The determinant on flat conic surfaces with excision of disks

Abstract:

Let $M$ be a surface with conical singularities, and consider a family of surfaces $M_{\epsilon }$ obtained from $M$ by removing disks of radius $\epsilon$ around a subset of the conical singularities. Such families arise naturally in the study of the moduli space of flat metrics on higher-genus surfaces with boundary. In particular, they have been used by Khuri to prove that the determinant of the Laplacian is not a proper map on this moduli space when the genus $p\geq 1$. Khuri’s work is closely related to the isospectral compactness results of Osgood, Phillips, and Sarnak. Our main theorem is an asymptotic formula for the determinant of $M_\epsilon$ as $\epsilon$ approaches zero up to terms which vanish in the limit. The proof uses the determinant gluing formula of Burghelea, Friedlander, and Kappeler along with an observation of Wentworth on the asymptotics of Dirichlet-to-Neumann operators. We then apply this theorem to extend and sharpen the results of Khuri.